Carbon-induced strengthening of bcc iron at the atomic scale

In steels, the interaction between screw dislocations and carbon solutes has a great influence on the yield strength. Fe-C potentials used in molecular dynamics (MD) simulations yield a poor description of screw dislocation properties—their core structure and Peierls barrier—compared to ab initio calculations. Here we combine two EAM potentials from the literature, which greatly improves dislocation property accuracy in FeC alloys. Using this hybrid potential, MD simulations of dislocation glide in random solid solutions confirm a powerful solute strengthening, caused by complex interaction processes. We analyze these processes in a model geometry, where a row of carbon atoms is inserted in the dislocation core with varying separations. We use a combination of MD simulations, minimum-energy path calculations, and a statistical model based on the harmonic transition state theory to explain the strengthening induced by carbon. We unveil that carbon disrupts the glide process, as unpinning requires the successive nucleation of two kink pairs. When solute separation is below about 100 Burgers vectors, the activation enthalpy of both kink pairs are markedly increased compared to pure iron, resulting in a strong dependence of the unpinning stress on solute spacing. Our simulations also suggest an effect of carbon spacing on the kink-pair activation entropy. This work provides elementary processes and parameters that will be useful for larger-scale models and, in particular, kinetic Monte Carlo simulations.

Figure 1

Differential displacements of atoms in the dislocation core region. Different structures were obtained: (a) compact core (DFT, all potentials except Lee and Henriksson); (b) degenerate core (Lee); (c) symmetrical hard core (DFT); and (d) asymmetrical hard core obtained with the hybrid and other potentials (see text). Grey circles represent the relaxed position of Fe atoms. Red circles represent carbon atoms. Head-up triangles correspond to easy positions, while head-down triangles correspond to hard positions.

Figure 2

Peierls barriers obtained with different interatomic potentials (blue dots) and compared to ab initio data [19] (red squares).

Figure 3

Carbon-dislocation binding energy obtained with different interatomic potentials and for different carbon spacings $d_{\mathrm{CC}}$ (represented by different symbols) and compared to DFT data [24]. Horizontal lines serve as guides for the eye.

Figure 4

Dislocation line (green) adopting a complex three-dimensional kinked structure in presence of carbon atoms (red spheres). Only the solutes closest to the dislocation are represented for clarity. The dislocation line was identified using the dxa algorithm [74].

Figure 5

Constant shear rate simulations with a carbon-depleted zone around the glide plane. (a) Stress-strain curves obtained at $300\mathrm{K}$. Blue lines represent simulations in which a carbon depleted zone of $n_{\mathrm{planes}}$ atomic planes above and below the glide plane was created, as illustrated in (b). The case $n_{\mathrm{planes}}=0$ corresponds to a full random solid solution with a carbon concentration equal to 0.52 at%. In the case $n_{\mathrm{planes}}=2$, the dislocation cross slips to the solid solution due to the attraction of solutes, resulting in a stress curve that is similar to the full solid solution case. The gray line in (a) represents the same deformation test conducted in pure Fe.

Figure 6

Local conversion of the screw dislocation core during a dynamical simulation. The smaller red atom is a carbon atom, while all others are iron atoms. Iron atoms that adopt a distorted prismatic structure near the carbon atom, are highlighted in blue. The dislocation line obtained using the dxa algorithm is shown in green. For clarity, only core atoms are shown.

Figure 7

Unpinning process for a dislocation pinned by two carbon atoms. (a) Snapshots taken at typical stages. The dislocation line as identified by the dxa algorithm is in green, carbon atoms are red spheres. Dashed lines serve as a visual guide. Between steps 3 and 4, a kink pair nucleates on the long segment, expands until being blocked by the carbon solutes, and spontaneously induces nucleation on the short segment. (b) Corresponding stress-strain curve. Numbers in (b) refer to the images in (a). Carbon atoms are separated by $\ell =10b$ along a $40b$ dislocation. The MD simulation was conducted at $40\mathrm{K}$ for $2\mathrm{n}\mathrm{s}$ with a strain rate of $1\times {10}^7{\mathrm{s}}^{-1}$.

Figure 8

Effect of the random initial velocity on the unpinning stress. Stress-strain curves obtained with C atoms every 20 $b$ along the dislocation segment, at $300\mathrm{K}$. Twenty different random seeds are used for the initial atomic velocities. The unpinning stress ${\tau }^{*}$ is marked by orange circles.

Figure 9

Dependence of the average unpinning stress on the largest dislocation segment length $L$, computed during MD simulations at (a) 100 and (b) 300 K. Blue circles are when $\ell <L$, red squares when $\ell =L$. Half-filled orange symbols refer to the flow stress in pure iron computed by MD in cells of different lengths $L$. The dashed and solid lines are predictions of a statistical model (see text for details).

Figure 10

NEB calculations of the two-step unpinning process of a dislocation from a row of C atoms. (a) Successive metastable and activated states during the unpinning process for a segment length of 30b and an applied stress of $600\mathrm{M}\mathrm{Pa}$. Core atoms were identified using the centrosymmetry parameter [75]. (b) Corresponding energy profile. (c) and (d) show the activation enthalpies of both transitions $\mathrm{\Delta }{H}_1$ and $\mathrm{\Delta }{H}_2$ (circles) for two different solute separations. The solid lines are fits using Kocks' law [Eq. (9)]. (e) and (f) show the fitted parameters $\mathrm{\Delta }{H}_{0,i}$ and ${\tau }_{P,i}$ of Kocks' law. Solid lines in (e) and (f) represent the average of $\mathrm{\Delta }{H}_0$ and a fit of ${\tau }_P$ using Eq. (10) for both transitions. Dashed lines in (c) to (f) represent the corresponding quantities in pure iron.